TITLE:
Renormalized Solutions of a Nonlinear Elliptic-Parabolic Equation with L1 Data and Finite-Element Numerical Simulation
AUTHORS:
Thierno Sarr, Souleye Kane
KEYWORDS:
Renormalized Solutions, Elliptic-Parabolic Equation, Porous Media, Richards Equation, Finite Elements, Picard Iteration, Data, Semigroup Theory
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.6,
June
16,
2026
ABSTRACT: We study the existence and uniqueness of renormalized solutions for a nonlinear degenerate elliptic-parabolic problem of the form
?M(
u
)
?t
?div(
?u+K(
u
)
e
z
)=f??in?
Q
T
=(
0,T
)×Ω,
with
L
1
data
f∈
L
1
(
Q
T
)
and
u
0
∈
L
1
(
Ω
)
, arising from the Richards model of unsaturated flow in porous media. Because classical weak solutions fail to be unique at this low-regularity level, we work within the framework of renormalized solutions, which select the physically relevant solution through an energy-dissipation condition at infinity. Existence is established via a double approximation scheme
(
ψ
m,n
)
combined with nonlinear semigroup theory, and uniqueness follows from a comparison principle obtained by the doubling-of-variables technique. We further present a conforming finite-element /implicit-Euler/Picard-iteration scheme validated on the Brooks-Corey soil model, for which we prove an
O(
h
2
+τ
)
error estimate in
L
2
(
Ω
)
and unconditional stability.